Methods and apparatus for sub-block based architecture of cholesky decomposition and channel whitening

ABSTRACT

Methods and apparatus for sub-block based architecture of Cholesky decomposition and channel whitening. In an exemplary embodiment, an apparatus is provided that parallel processes sub-block matrices (R 00 , R 10 , and R 11 ) of a covariance matrix (R) to determine a whitening coefficient matrix (W). The apparatus includes a first LDL coefficient calculator that calculates a first whitening matrix W 00 , lower triangle matrix L 00 , and diagonal matrix D 00  from the sub-block matrix R 00 , a first matrix calculator that calculates a lower triangle matrix L 10  from the sub-block matrix R 10  and the matrices L 00  and D 00 , and a second matrix calculator that calculates a matrix X from the matrices D 00  and L 10 . The apparatus also includes a matrix subtractor that calculates a matrix Z from the matrix X and the sub-block matrix R 11 , a second LDL coefficient calculator that calculates a third whitening matrix W 11 , lower triangle matrix L 11 , and a diagonal matrix D 11  from the matrix Z, and a third matrix calculator that calculates a second whitening matrix W 10  from the matrices L 00 , L 10 , L 11 , and D 11 .

CLAIM TO PRIORITY

This application claims the benefit of priority from U.S. ProvisionalApplication No. 62/665,271, filed on May 1, 2018, and entitled “METHODAND APPARATUS FOR SUB-BLOCK BASED ARCHITECTURE OF CHOLESKY DECOMPOSITIONAND CHANNEL WHITENING,” which is incorporated by reference herein in itsentirety.

FIELD

The exemplary embodiment(s) of the present invention relates totelecommunications network. More specifically, the exemplaryembodiment(s) of the present invention relates to receiving andprocessing data stream via a wireless communication network.

BACKGROUND

With a rapidly growing trend of mobile and remote data access over ahigh-speed communication network such as third (3G), fourth (4G) orfifth (5G) generation cellular services, accurately delivering anddeciphering data streams become increasingly challenging and difficult.The high-speed communication network which is capable of deliveringinformation includes, but not limited to, wireless network, cellularnetwork, wireless personal area network (“WPAN”), wireless local areanetwork (“WLAN”), wireless metropolitan area network (“MAN”), or thelike. While WPAN can be Bluetooth or ZigBee, WLAN may be a Wi-Fi networkin accordance with IEEE 802.11 WLAN standards.

In the Long-Term Evolution (LTE) or 5G standard, pre-whitening isapplied in the receiver to whiten a correlated noise covariance matrixto be un-correlated to ease signal detection. Typically, the whiteningis done by using a Cholesky Decomposition, such as a well-known LDL orLL decomposition procedure to generate whitening filters. However, theoriginal LDL decomposition involves an iterative procedure depending onthe dimension of the input matrix. Also, its sequential nature does notfacilitate easy parallelization of the computation for efficient VLSIimplementations, especially when the dimension of the input matrixgrows.

Therefore, it is desirable to have a way to perform channel whiteningthat overcomes problems associated with iterative processes and thatfacilitate easy parallelization of the computation for efficient VLSIimplementations, especially when the dimension of the input matrixgrows.

SUMMARY

The following summary illustrates a simplified version(s) of one or moreaspects of present invention. The purpose of this summary is to presentsome concepts in a simplified description as more detailed descriptionthat will be presented later.

In various exemplary embodiments, methods and apparatus are disclosedfor a sub-block architecture for Cholesky decomposition to providechannel whitening. In various exemplary embodiments, a covariance matrixis broken into sub-blocks to facilitate the parallel calculation ofwhitening coefficients. A specialized 2×2 coefficient calculator isprovided that receives a Hermitian matrix and calculates whiteningcoefficients. An apparatus is provided that utilizes the coefficientcalculator to process multiple sub-blocks of the covariance matrix inparallel to calculate a complete whitening coefficient matrix. Theapparatus including the specialized calculator perform channel whiteningusing a parallel process overcomes problems associated with iterativeprocesses, and that facilitate efficient VLSI implementations,especially when the dimension of the input matrix grows.

In an exemplary embodiment, an apparatus is provided that parallelprocesses sub-block matrices (R₀₀, R₁₀, and R₁₁) of a covariance matrix(R) to determine a whitening coefficient matrix (W). The apparatusincludes a first LDL coefficient calculator that calculates a firstwhitening matrix W₀₀, lower triangle matrix L₀₀, and diagonal matrix D₀₀from the sub-block matrix R₀₀, a first matrix calculator that calculatesa lower triangle matrix L₁₀ from the sub-block matrix R₁₀ and thematrices L₀₀ and D₀₀, and a second matrix calculator that calculates amatrix X from the matrices D₀₀ and L₁₀. The apparatus also includes amatrix subtractor that calculates a matrix Z from the matrix X and thesub-block matrix R₁₁, a second LDL coefficient calculator thatcalculates a third whitening matrix W₁₁, lower triangle matrix L₁₁, anda diagonal matrix D₁₁ from the matrix Z, and a third matrix calculatorthat calculates a second whitening matrix W₁₀ from the matrices L₀₀,L₁₀, L₁₁, and D₁₁.

In an exemplary embodiment, a method is disclosed for parallelprocessing sub-block matrices (R₀₀, R₁₀, and R₁₁) of a covariance matrix(R) to determine a whitening coefficient matrix (W). The method includescalculating a first whitening matrix W₀₀, lower triangle matrix L₀₀, anddiagonal matrix D₀₀ from the sub-block matrix R₀₀, calculating a lowertriangle matrix L₁₀ from the sub-block matrix R₁₀ and the matrices L₀₀and D₀₀, and calculating a matrix X from the matrices D₀₀ and L₁₀. Themethod also includes calculating a matrix Z from the matrix X and thesub-block matrix R₁₁, calculating a third whitening matrix W₁₁, lowertriangle matrix L₁₁, and a diagonal matrix D₁₁ from the matrix Z, andcalculating a second whitening matrix W₁₀ from the matrices L₀₀, L₁₀,L₁₁, and D₁₁

Additional features and benefits of the exemplary embodiment(s) of thepresent invention will become apparent from the detailed description,figures and claims set forth below.

BRIEF DESCRIPTION OF THE DRAWINGS

The exemplary aspect(s) of the present invention will be understood morefully from the detailed description given below and from theaccompanying drawings of various embodiments of the invention, which,however, should not be taken to limit the invention to the specificembodiments, but are for explanation and understanding only.

FIG. 1 is a block diagram illustrating a communication networkconfigured to transmit and receive data streams using variousembodiments of a whitening coefficient calculator to provide channelwhitening;

FIG. 2 is a block diagram illustrating an LTE/5G PUCCH receiver thatincludes an exemplary embodiment of a whitening coefficient calculatorthat generates whitening coefficients for channel whitening;

FIG. 3 shows an exemplary framework for use in generating channelwhitening coefficients;

FIG. 4 is a block diagram illustrating an exemplary embodiment of a 2×2LDL and whitening coefficient calculator;

FIG. 5 is a block diagram of a detailed exemplary embodiment of the 2×2LDL and whitening coefficient calculator shown in FIG. 4;

FIG. 6 is a block diagram illustrating an alternative exemplaryembodiment of a D/L calculator;

FIG. 7 is a block diagram illustrating an exemplary embodiment of a 4×4matrix LDL decomposition and whitening matrix generator based on a 2×2sub-block decomposition in accordance with exemplary embodiments;

FIG. 8 shows an exemplary embodiment of a method for determine whiteningcoefficients using a sub-block Cholesky architecture; and

FIG. 9 illustrates an exemplary digital computing system with variousfeatures for network communication that include a sub-blockdecomposition and whitening coefficient calculator and associatedmethods.

DETAILED DESCRIPTION

Aspects of the present invention are described herein comprising methodsand apparatus for a sub-block architecture for Cholesky decomposition toprovide channel whitening.

The purpose of the following detailed description is to provide anunderstanding of one or more embodiments of the present invention. Thoseof ordinary skills in the art will realize that the following detaileddescription is illustrative only and is not intended to be in any waylimiting. Other embodiments will readily suggest themselves to suchskilled persons having the benefit of this disclosure and/ordescription.

In the interest of clarity, not all of the routine features of theimplementations described herein are shown and described. It will, ofcourse, be understood that in the development of any such actualimplementation, numerous implementation-specific decisions may be madein order to achieve the developer's specific goals, such as compliancewith application- and business-related constraints, and that thesespecific goals will vary from one implementation to another and from onedeveloper to another. Moreover, it will be understood that such adevelopment effort might be complex and time-consuming but wouldnevertheless be a routine undertaking of engineering for those ofordinary skills in the art having the benefit of embodiment(s) of thisdisclosure.

Various embodiments of the present invention illustrated in the drawingsmay not be drawn to scale. Rather, the dimensions of the variousfeatures may be expanded or reduced for clarity. In addition, some ofthe drawings may be simplified for clarity. Thus, the drawings may notdepict all of the components of a given apparatus (e.g., device) ormethod. The same reference indicators will be used throughout thedrawings and the following detailed description to refer to the same orlike parts.

The term “system” or “device” is used generically herein to describe anynumber of components, elements, sub-systems, devices, packet switchelements, packet switches, access switches, routers, networks, modems,base stations, eNB (eNodeB), computer and/or communication devices ormechanisms, or combinations of components thereof. The term “computer”includes a processor, memory, and buses capable of executing instructionwherein the computer refers to one or a cluster of computers, personalcomputers, workstations, mainframes, or combinations of computersthereof.

IP communication network, IP network, or communication network means anytype of network having an access network that is able to transmit datain a form of packets or cells, such as ATM (Asynchronous Transfer Mode)type, on a transport medium, for example, the TCP/IP or UDP/IP type. ATMcells are the result of decomposition (or segmentation) of packets ofdata, IP type, and those packets (here IP packets) comprise an IPheader, a header specific to the transport medium (for example UDP orTCP) and payload data. The IP network may also include a satellitenetwork, a DVB-RCS (Digital Video Broadcasting-Return Channel System)network, providing Internet access via satellite, or an SDMB (SatelliteDigital Multimedia Broadcast) network, a terrestrial network, a cable(xDSL) network or a mobile or cellular network (GPRS/EDGE, or UMTS(where applicable of the MBMS (Multimedia Broadcast/Multicast Services)type, or the evolution of the UMTS known as LTE (Long Term Evolution),or DVB-H (Digital Video Broadcasting-Handhelds)), or a hybrid (satelliteand terrestrial) network.

FIG. 1 is a block diagram illustrating a communication network 100configured to transmit and receive data streams using variousembodiments of a sub-block architecture for Cholesky decomposition toprovide channel whitening. The network 100 includes packet data networkgateway (“P-GW”) 120, two serving gateways (“S-GWs”) 121-122, two basestations (or cell sites) 102-104, server 124, and Internet 150. P-GW 120includes various components 140 such as billing module 142, subscribingmodule 144, tracking module 146, and the like to facilitate routingactivities between sources and destinations. It should be noted that theunderlying concept of the exemplary embodiment(s) of the presentinvention would not change if one or more blocks (or devices) were addedto or removed from diagram 100.

The network configuration illustrated by the communication network 100may also be referred to as a third generation (“3G”), 4G, LTE, 5G, orcombination of 3G and 4G cellular network configuration. MME 126, in oneaspect, is coupled to base stations (or cell site) and S-GWs capable offacilitating data transfer between 3G and LTE or between 2G and LTE. MME126 performs various controlling/managing functions, network securities,and resource allocations.

S-GW 121 or 122, in one example, coupled to P-GW 120, MME 126, and basestations 102 or 104, is capable of routing data packets from basestation 102, or eNodeB, to P-GW 120 and/or MME 126. A function of S-GW121 or 122 is to perform an anchoring function for mobility between 3Gand 4G equipment. S-GW 122 is also able to perform various networkmanagement functions, such as terminating paths, paging idle UEs,storing data, routing information, generating replica, and the like.

P-GW 120, coupled to S-GWs 121-122 and Internet 150, is able to providenetwork communication between user equipment (“UE”) and IP basednetworks such as Internet 150. P-GW 120 is used for connectivity, packetfiltering, inspection, data usage, billing, or PCRF (policy and chargingrules function) enforcement, et cetera. P-GW 120 also provides ananchoring function for mobility between 3G and 4G (or LTE) packet corenetwork(s).

Sectors or blocks 102-104 are coupled to a base station or FEAB 128,which may also be known as cell site, node B, or eNodeB. Sectors 102-104include one or more radio towers 110 or 112. Radio tower 110 or 112 isfurther coupled to various UEs, such as a cellular phone 106, a handhelddevice 108, tablets and/or iPad® 107 via wireless communications orchannels 137-139. Devices 106-108 can be portable devices or mobiledevices, such as iPhone®, BlackBerry®, Android®, and so on. Base station102 facilitates network communication between mobile devices such as UEs106-107 with S-GW 121 via radio towers 110. It should be noted that basestation or cell site can include additional radio towers as well asother land switching circuitry.

Server 124 is coupled to P-GW 120 and base stations 102-104 via S-GW 121or 122. In one embodiment, server 124 which contains a soft decodingscheme 128 is able to distribute and/or manage soft decoding and/or harddecoding based on predefined user selections. In one exemplary instance,upon detecting a downstream push data 130 addressing to mobile device106 which is located in a busy traffic area or noisy location, basestation 102 can elect to decode the downstream using the soft decodingscheme distributed by server 124. One advantage of using the softdecoding scheme is that it provides more accurate data decoding, wherebyoverall data integrity may be enhanced.

When receiving bit-streams via one or more wireless or cellularchannels, a decoder can optionally receive or decipher bit-streams withhard decision or soft decision. A hard decision is either 1 or 0 whichmeans any analog value greater than 0.5 is a logic value one (1) and anyanalog value less than 0.5 is a logic value zero (0). Alternatively, asoft decision or soft information can provide a range of value from 0,0.2, 0.4, 0.5, 0.6, 0.8, 0.9, and the like. For example, softinformation of 0.8 would be deciphered as a highly likelihood one (1)whereas soft information of 0.4 would be interpreted as a weak zero (0)and maybe one (1).

A base station, in one aspect, includes one or more FEABs 128. Forexample, FEAB 128 can be a transceiver of a base station or eNodeB. Inone aspect, mobile devices such tables or iPad® 107 uses a first type ofRF signals to communicate with radio tower 110 at sector 102 andportable device 108 uses a second type of RF signals to communicate withradio tower 112 at sector 104. After receiving RF samples, FEAB 128 isable to process samples using a whitening coefficient calculator (WCC)152 that generates whitening coefficients to implement a whiteningfilter. In an exemplary embodiment, the WCC 152 determines whiteningcoefficients using a sub-block Cholesky decomposition architecture thatovercomes problems associated with iterative processes and facilitateseasy parallelization of the computation for efficient VLSIimplementations, especially when the dimension of the input matrixgrows.

FIG. 2 is a block diagram illustrating an LTE/5G PUCCH receiver 200 thatincludes an exemplary embodiment of a whitening coefficient calculator206 that generates whitening coefficients for channel whitening. Forexample, the receiver 200 is suitable for use in one or more of thecommunicating devices shown in FIG. 1.

The receiver 200 comprises a front-end receiver 202 that providesfrequency domain processes to received RF signals and outputs datasymbols to a data symbol processor 204 and pilot symbols to a pilotsymbol processor 208. The data symbol processor 204 performs DFTfunctions that include data whitening. The pilot symbol processor 208performs pilot AFC and pilot whitening. Both the data symbol processor204 and the pilot symbol processor 208 provide inputs to the WCC 206 toobtain whitening coefficients that can be used to implement whiteningfilters. In various exemplary embodiments, the WCC 206 performs 2×2sub-block Cholesky decomposition to generate whitening coefficients thatare passed to the data symbol processor 204 and pilot symbol processor208, which apply whitening filters to the data and pilot information.

The data symbol processor 204 and pilot symbol processor 208 outputwhitened data and pilot information to a data channel compensator 210and a pilot channel estimator 212, respectively. The pilot channelestimator 212 output channel estimates to the data channel compensator210. After the data channel compensator 210, the data flows to adiversity MRC 214, demodulator/descrambler 216, and decode bit demapper218 that outputs data for further processing.

In various exemplary embodiments, the WCC 206 operates to divide aCholesky decomposition into sub-blocks that allow the whiteningcoefficients to be determined with greater efficiency than conventionsystems. A more detailed description of the operation of the WCC 206 isprovided below.

Whitening Coefficient Calculator

In various exemplary embodiments, a whitening filter coefficientcalculator performs a sub-block Cholesky (LDL) decomposition for ahigher order multiple input and multiple output (MIMO) system. In anexemplary embodiment, a 4×4 MIMO matrix decomposition is built upon asimpler computation module of 2×2 dimension. This is achieved bybreaking the 4×4 matrix into a 2×2 matrix of 2×2 sub-blocks. Theresulting architecture is more efficient for VLSI implementation andmore modular than conventional architectures.

Whitening involves the following matrix computations. Assume a noisecovariance matrix of dimension N×N, where N is the number of receiveantennas in a MIMO system. In an exemplary embodiment, the followingcovariance matrix (Ruu), lower triangle matrix (L) and diagonal matrix(D) can be defined.

R_(uu) = LDL^(H) ${L = \begin{bmatrix}1 & \; & \; & 0 \\l_{10} & 1 & \; & \; \\\ldots & \; & \ldots & \; \\l_{{N - 1},0} & \; & l_{{N - 1},{N - 2}} & 1\end{bmatrix}},{D = {{diag}\left( {d_{0,0},\ldots \mspace{14mu},d_{{N - 1},{N - 1}}} \right)}}$

A whitening filter can be designed as W=(D)^(1/2)L⁻¹ and can be appliedto filter the received signal to generate a whitened signal (r_(whiten))from the product of the whitening filter (W) and the received inputvalues (r_(in)) as follows.

r _(whiten) =W ^(H) r _(in)

FIG. 3 shows an exemplary architecture 300 that comprises a WCC 316 thatutilizes an LDL decomposition. The architecture 300 comprises an RFfront end 302, noise covariance estimator 304, WCC 316 and whiteningfilter 310. The WCC 316 comprise LDL decomposition circuit 306 andwhitening filter coefficient generator 308. During operation, thereceived signal (r_(in)) is input to the noise covariance estimator 304and the whitening filter 310. The noise covariance estimator 304 outputsa noise covariance matrix (Ruu) that is input to the LDL decompositioncircuit 306. The decomposition circuit 306 generates the lower triangle(L) and diagonal (D) matrices as described above. These matrices areinput to the whitening filter coefficient generator 308 that generatesthe whitening coefficients (W). In an exemplary embodiment, thedecomposition is divided into sub-blocks to enhance efficiency andsimplify implementation. The whitening filter 310 applies the whiteningfilter coefficients (W) to the input (r_(in)) to generate the whitenedoutput (r_(whiten)) 314.

Whitening Coefficient Calculator using a 2×2 LDL Decomposition

In an exemplary embodiment, the case of a 2×2 covariance matrix case isdefined as follows.

${R_{uu} = \begin{bmatrix}r_{00} & r_{01} \\r_{10} & r_{11}\end{bmatrix}},{L = \begin{bmatrix}1 & \; \\l_{10} & 1\end{bmatrix}},{D = {{diag}\left( {d_{00},d_{11}} \right)}}$

where LDL^(H)=R_(uu). A method to compute whitening matrix coefficientsis described as follows. Computations of diagonal (d₀₀, d₁₁) and lowertriangle matrix (l₁₀) components can be expressed as illustrated in thefollowing expressions.

d₀₀ = r₀₀$d_{11} = {{r_{11} - \frac{{r_{10}}^{2}}{r_{00}}} = {r_{11} - \left( \frac{{r_{10 \cdot {Re}}*r_{10 \cdot {Re}}} + {r_{10 \cdot {Im}}*r_{10 \cdot {Im}}}}{r_{00}} \right)}}$$l_{1} = {\frac{r_{10}}{r_{00}} = {\frac{r_{10 \cdot {Re}} + {jr}_{10 \cdot {Im}}}{r_{0}}.}}$

while the whitening matrix coefficients can be computed from the fourequations as illustrated in the following expressions.

W ₀₀=1/sqrt(d ₀₀)

W ₀₁=0

W ₁₀ =−l ₁₀/sqrt(d ₁₁)

W ₁₁=1/sqrt(d ₁₁)

FIG. 4 is a block diagram illustrating an exemplary embodiment of a 2×2sub-block LDL WCC 400. For example, the WCC 400 is suitable for use withthe WCC 316 shown in FIG. 3 and the WCC 206 shown in FIG. 2. In anexemplary embodiment, the WCC 400 implements the 2×2 LDL decompositionwith only independent coefficients. The WCC 400 receives covariancematrix elements 402 and processes these elements to determine D, L, andW matrix values 404. A more detailed description of the WCC 400 isprovided below.

FIG. 5 is a block diagram of a detailed exemplary embodiment of a 2×2sub-block LDL WCC 500. For example, the WCC 500 is suitable for use asthe WCC 400 shown in FIG. 4. The WCC 500 comprises D/L calculator 502and coefficient calculator 504. In an exemplary embodiment, the D/Lcalculator 502 receives matrix elements of a Hermitian covariance 2×2matrix and computes D and L matrix values. For example, the D/Lcalculator 502 uses five multipliers, two adders, and a (1/x) look-uptable (LUT) 506 to compute the desired D 512 and L 514 matrix values asillustrated in the above expressions.

The outputs from the D/L calculator 502 are input to the coefficientcalculator 504 that operates to calculate whitening filter coefficients510. For example, the coefficient calculator 504 uses two multipliersand two (1/sqrt (x)) (LUTs) 508, 516 to compute the desired whiteningcoefficients 510 from the outputs of the D/L calculator 502, asillustrated in the above expressions.

FIG. 6 is a block diagram illustrating an alternative exemplaryembodiment of a D/L calculator 600. For example, the D/L calculator 600is suitable to replace the D/L calculator 502 shown in FIG. 5. In anexemplary embodiment, the D/L calculator 600 comprises four multipliers,two adders, and a (1/x) look-up table (LUT) 506 to compute the desired D512 and L 514 matrix values to provide a variation with a more efficientdata-path and whose computation equations are illustrated in thefollowing expressions.

d₀₀ = r₀₀$l_{10} = {\frac{r_{10 \cdot {Re}}}{r_{00}} + {j\frac{r_{10 \cdot {Im}}}{r_{00}}}}$$d_{11} = {r_{11} - \left( {{\frac{r_{10 \cdot {Re}}}{r_{00}}*r_{10 \cdot {Re}}} + {\frac{r_{10 \cdot {Im}}}{r_{00}}*r_{10 \cdot {Im}}}} \right)}$

Example: Generating Whitening Coefficients from LDL Sub-BlockDecomposition

The following is an example illustrating how the WCC performs a 4×4 LDLsub-block decomposition to calculate whitening coefficients inaccordance with the exemplary embodiments. A 4×4 covariance matrix(R_(4×4)) is broken down into 2×2 matrices of 2×2 sub-blocks asillustrated in the following expressions.

$R_{4 \times 4} = {\begin{bmatrix}r_{00} & r_{01} & r_{02} & r_{03} \\r_{10} & r_{11} & r_{12} & r_{13} \\r_{20} & r_{21} & r_{22} & r_{23} \\r_{30} & r_{31} & r_{32} & r_{33}\end{bmatrix} = \begin{bmatrix}\begin{bmatrix}r_{00} & r_{01} \\r_{10} & r_{11}\end{bmatrix} & \begin{bmatrix}r_{20}^{*} & r_{30}^{*} \\r_{21}^{*} & r_{31}^{*}\end{bmatrix} \\\begin{bmatrix}r_{20} & r_{21} \\r_{30} & r_{31}\end{bmatrix} & \begin{bmatrix}r_{22} & r_{32}^{*} \\r_{32} & r_{33}\end{bmatrix}\end{bmatrix}}$ $R_{00} = \begin{bmatrix}r_{00} & r_{01} \\r_{10} & r_{11}\end{bmatrix}$ $R_{10} = \begin{bmatrix}r_{20} & r_{21} \\r_{30} & r_{31}\end{bmatrix}$ $R_{11} = \begin{bmatrix}r_{22} & r_{32}^{*} \\r_{32} & r_{33}\end{bmatrix}$

Therefore, from the above,

$R_{4 \times 4} = {\begin{bmatrix}R_{00} & \left( R_{10} \right)^{H} \\R_{10} & R_{11}\end{bmatrix}.}$

The LDL decomposition can be broken down into the LDL decomposition ofdimension 2×2 as illustrated in the following expressions.

$R = {\begin{bmatrix}R_{00} & R_{10}^{H} \\R_{10} & R_{11}\end{bmatrix} = {LDL}^{H}}$ ${L = \begin{bmatrix}L_{00} & 0 \\L_{10} & L_{11}\end{bmatrix}},\mspace{11mu} {D = \begin{bmatrix}D_{00} & 0 \\0 & D_{11}\end{bmatrix}},{L_{00} = \begin{bmatrix}1 & 0 \\l_{00}^{(10)} & 1\end{bmatrix}},\mspace{11mu} {D_{00} = \begin{bmatrix}d_{00}^{(00)} & 0 \\0 & d_{00}^{(11)}\end{bmatrix}},\mspace{11mu} {D_{11} = \begin{bmatrix}d_{11}^{(00)} & 0 \\0 & d_{11}^{(11)}\end{bmatrix}}$ ${L_{11} = \begin{bmatrix}1 & 0 \\l_{11}^{(10)} & 1\end{bmatrix}},{L_{10} = {\begin{bmatrix}l_{10}^{(00)} & l_{10}^{(01)} \\l_{10}^{(10)} & l_{10}^{(11)}\end{bmatrix}.}}$

Given the above, the 4×4 LDL decomposition is solved with the followingprocedure using the 2×2 modules shown in FIGS. 5-6, where the operator[ldl_r2( )] performs the operations of the WCC 500 on the identifiedinput 2×2 matrix, for example, as described with reference to FIG. 5.

$\mspace{20mu} {{\begin{bmatrix}R_{00} & R_{10}^{H} \\R_{10} & R_{11}\end{bmatrix} = \begin{bmatrix}{L_{00}D_{00}L_{00}^{H}} & \left( {L_{10}D_{00}L_{00}^{H}} \right)^{H} \\{L_{10}D_{00}L_{00}^{H}} & {{L_{10}D_{00}L_{10}^{H}} - {L_{11}D_{11}L_{11}^{H}}}\end{bmatrix}},\left\{ {\begin{matrix}{{R_{00} = {\left. {L_{00}D_{00}L_{00}^{H}}\Rightarrow\left( {L_{00},D_{00}} \right) \right. = {{ldl\_ r}\; 2\left( R_{00} \right)}}};} \\{{L_{10}D_{00}L_{00}^{H}} = {\left. R_{10}\Rightarrow L_{10} \right. = {R_{10}\left( {D_{00}L_{00}^{H}} \right)}^{- 1}}} \\{{L_{11}D_{11}L_{11}^{H}} = {\left. {R_{11} - {L_{10}D_{00}L_{10}^{H}}}\Rightarrow\left( {L_{11},D_{11}} \right) \right. = {{ldl\_ r2}\left( {R_{11} - {L_{10}D_{00}L_{10}^{H}}} \right)}}}\end{matrix};} \right.}$

The whitening filter coefficients can be solved as illustrated in thefollowing expressions.

$\mspace{20mu} {W = {{D^{- \frac{1}{2}}(L)}^{- 1} = \begin{bmatrix}W_{00} & 0 \\W_{10} & W_{11}\end{bmatrix}}}$ $\mspace{20mu} {(L)^{- 1} = \begin{bmatrix}L_{00}^{- 1} & 0 \\{{- L_{11}^{- 1}}L_{10}L_{00}^{- 1}} & L_{11}^{- 1}\end{bmatrix}}$$\mspace{20mu} {W_{10} = {{- {D_{11}^{{- 1}/2}\left( {L_{11}^{- 1}L_{10}L_{00}^{- 1}} \right)}} = {- \begin{bmatrix}\frac{m_{00}}{\sqrt{d_{11}^{(00)}}} & \frac{m_{01}}{\sqrt{d_{11}^{(00)}}} \\\frac{m_{10}}{\sqrt{d_{11}^{(00)}}} & \frac{m_{11}}{\sqrt{d_{11}^{(00)}}}\end{bmatrix}}}}$   where$\mspace{20mu} {M = {{L_{11}^{- 1}L_{10}L_{00}^{- 1}} = \begin{bmatrix}m_{00} & m_{01} \\m_{10} & m_{11}\end{bmatrix}}}$ $Q = {{L_{11}^{- 1}L_{10}} = \begin{bmatrix}{L_{10}\left( {0,0} \right)} & {L_{10}\left( {0,1} \right)} \\\begin{matrix}{{L_{10}\left( {1,0} \right)} - {{L_{11}\left( {1,0} \right)}*}} \\{L_{10}\left( {0,0} \right)}\end{matrix} & {{L_{10}\left( {1,1} \right)} - {{L_{11}\left( {1,0} \right)}*{L_{10}\left( {0,1} \right)}}}\end{bmatrix}}$ $\mspace{20mu} \left\{ \begin{matrix}{m_{00} = {{L_{10}\left( {0,0} \right)} - {{L_{10}\left( {0,1} \right)}*{L_{00}\left( {1,0} \right)}}}} \\{m_{01} = {L_{10}\left( {0,1} \right)}} \\{m_{10} = {q_{10} - {{L_{00}\left( {1,0} \right)}q_{11}}}} \\{m_{11} = q_{11}}\end{matrix} \right.$

FIG. 7 is a block diagram illustrating an exemplary embodiment of a 4×4matrix LDL decomposition and whitening coefficient calculator 700 basedon a 2×2 sub-block decomposition in accordance with exemplaryembodiments. The calculator 700 provides computation for both the LDLdecomposition and the whitening filter coefficient matrix.

The calculator 700 utilizes a 2×2 LDL WCC 702 (e.g., shown in FIGS. 4-6)as a simple base module for the 2×2 matrix decomposition and another 2×2LDL WCC 704 when the matrix dimension grows to 4×4.

In an exemplary embodiment, a sub-block generator 730 generates 2×2sub-blocks from an input covariance matrix. A first 2×2 LDL WCC 702generates (L₀₀, D₀₀, and W₀₀) from the covariance matrix sub-block(R₀₀). A second 2×2 LDL WCC 704 generates (L₁₁, D₁₁, and W₁₁) from thecovariance matrix sub-block (R₁₁) minus [Z=R₁₁−X]. For example, thegenerators 702 and 704 perform the expressions shown above.

The calculator 700 also comprises L10 matrix calculator 706, matrixmultiplier 708, and matrix subtractor 710. The L10 matrix calculator 706includes inversion calculator 716 and matrix multiplier 718.

During operation, the 2×2 LDL WCC 702 receives the R₀₀ matrix andcalculates the matrices L₀₀, D₀₀, and W₀₀ as described in the aboveexpressions. The inversion function 716 of the L10 matrix calculator 706receives the L₀₀ and D₀₀ matrices and calculates the matrix (D₀₀L₀₀^(H))⁻¹ that is input to a matrix multiplier 718. The multiplier 718also receives the matrix R₁₀ and calculates the matrix L₁₀ fromR₁₀(D₀₀L₀₀ ^(H))⁻¹.

The matrix multiplier 708 receives the matrices D₀₀ and L₁₀ andcalculates a matrix X from L₀₀D₀₀L₁₀ ^(H). The matrix subtractor 710subtracts the matrix X from the matrix R₁₁ to generate a matrix (Z) thatis input to the 2×2 LDL WCC 704.

The W10 calculator 712 receives the L₁₀ matrix, the L₀₀ matrix, and theL₁₁ and D₁₁ matrices output from the 2×2 LDL WCC 704. The W10 calculator712 uses the received matrices to calculate W₁₀ by calculating theexpression −D₁ ^(1/2)(L₁₁ ⁻¹L₁₀L₀₀ ⁻¹).

An output matrix combiner block 714 is shown but is optional. Forexample, it may be desirable output individual matrix elements. However,in an embodiment, the block 714 receives the various calculated L, D,and W matrices and consolidates the sub-block results into large 4×4output matrices. For example, 4×4 matrices for L and D 724 are formed bycombiner 720 and a 4×4 matrix for W 726 (whitening coefficients) isformed by combiner 722. These matrices can be utilized by otherfunctions to perform signal whitening or pilot whitening as illustratedin FIG. 2.

Compared with a conventional and sequential decomposition architecturethat is iterative on each increasing dimension, the novel sub-blockarchitecture described above provides significant improvements andadvantages, which include the following.

-   1. More scalable for the increasing dimension of 2×2 and 4×4    matrices.-   2. Reusable modules and data paths.-   3 More parallelism and efficient utilization of the data-path and    logic are better suited for high throughput VLSI implementations.

Accordingly, the various exemplary embodiments of the sub-blockdecomposition architecture generate whitening coefficients and can beused in various receiver processing flows, including but not limited toan LTE/NR PUCCH receiver or a PUSCH receiver. One example of the use ofthe disclosed sub-block architecture is shown in the PUCCH F3 receiver,as illustrated in FIG. 2.

FIG. 8 shows an exemplary embodiment of a method 800 for calculatingwhitening coefficients using embodiments of the sub-block Choleskydecomposition as described herein. For example, the method 800 issuitable for use with the calculator 700 shown in FIG. 7.

At block 802, input values are received. For example, in an embodiment,the input values are derived from received MIMO signals.

At block 804, an estimated covariance matrix is generated from the inputvalues. For example, a 4×4 estimated covariance matrix is generated asillustrated in the expressions above.

At block 806, as part of an LDL decomposition, the covariance matrix isbroken into sub-blocks. For example, the covariance matrix is brokeninto 2×2 sub-blocks by sub-block generator 730 shown in FIG. 7 and asillustrated in the expressions above.

At block 808, matrices for L₀₀, D₀₀, and W₀₀ are calculated fromcovariance sub-block R₀₀. In an exemplary embodiment, the 2×2 LDL 702performs this calculation. For example, the 2×2 LDL WCC 500 shown inFIG. 5 is suitable for use to perform this calculation.

At block 810, the matrices for L₀₀ and D₀₀ are used to calculate amatrix (D₀₀L₀₀ ^(H))⁻¹. In an exemplary embodiment, the inversionfunction 716 performs this calculation.

At block 812, the matrix for R₁₀ is used to calculate L₁₀. In anexemplary embodiment, the matrix L₁₀ is calculated from R₁₀(D₀₀L₀₀^(H))⁻¹. In an exemplary embodiment, the multiplier 718 performs thiscalculation.

At block 814, the matrices for D₀₀ and L₁₀ are used to calculate amatrix X. For example, the matrix X is calculated from L₁₀D₀₀L₀₀ ^(H).In an exemplary embodiment, the multiplier 708 performs thiscalculation.

At block 816, the matrices for R₁₁ and X are received and used tocalculate a difference matrix (Z=R₁₁−X). In an exemplary embodiment, thesubtractor 710 performs this calculation.

At block 818, the matrices L₁₁, D₁₁, and W₁₁ are calculated from thedifference matrix (Z). In an exemplary embodiment, the 2×2 LDL WCC 704receives the difference matrix (Z) and calculates the matrices L₁₁, D₁₁,and W₁₁.

At block 820, the matrices L₀₀, L₁₀, D₁₁, and L₁₁ are received and usedto calculate W₁₀. For example, W₁₀ is calculated as −D₁₁ ^(1/2)(L₁₁⁻¹L₁₀L₀₀ ⁻¹). In an exemplary embodiment, the W10 calculator 712performs this calculation.

At block 822, the calculated D, L and whitening coefficient matrix W areoutput either by individual components or by a combined large matrix.For example, the matrix combiner 714 performs matrix formation to outputlarge matrices.

At block 824, the input values are filtered using a whitening filterthat is generated using the calculated whitening coefficients.

Thus, the method 800 operates to utilize a sub-block Choleskydecomposition architecture to determine whitening coefficients. Itshould be noted that the operations of the method 800 may be added to,subtracted from, deleted, changes, rearranged or otherwise modifiedwithin the scope of the embodiments.

FIG. 9 illustrates an exemplary digital computing system 900 withvarious features for network communication that include a sub-blockdecomposition and whitening coefficient calculator and associatedmethods as described above. It will be apparent to those of ordinaryskill in the art that other alternative computer system architecturesmay also be employed.

Computer system 900 includes a processing unit 901, an interface bus912, and an input/output (“IO”) unit 920. Processing unit 901 includes aprocessor 902, main memory 904, system bus 910, static memory device906, bus control unit 905, and mass storage memory 907. Bus 910 is usedto transmit information between various components and processor 902 fordata processing. Processor 902 may be any of a wide variety ofgeneral-purpose processors, embedded processors, or microprocessors suchas ARM® embedded processors, Intel® Core™2 Duo, Core™2 Quad, Xeon®,Pentium™ microprocessor, AMD® family processors, MIPS® embeddedprocessors, or Power PC™ microprocessor.

Main memory 904, which may include multiple levels of cache memories,stores frequently used data and instructions. Main memory 904 may be RAM(random access memory), MRAM (magnetic RAM), or flash memory. Staticmemory 906 may be a ROM (read-only memory), which is coupled to bus 911,for storing static information and/or instructions. Bus control unit 905is coupled to buses 910-912 and controls which component, such as mainmemory 904 or processor 902, can use the bus. Mass storage memory 907may be a magnetic disk, solid-state drive (“SSD”), optical disk, harddisk drive, floppy disk, CD-ROM, and/or flash memories for storing largeamounts of data.

I/O unit 920, in one example, includes a display 921, keyboard 922,cursor control device 923, decoder 924, and communication device 925.Display device 921 may be a liquid crystal device, flat panel monitor,cathode ray tube (“CRT”), touch-screen display, or other suitabledisplay device. Display 921 projects or displays graphical images orwindows. Keyboard 922 can be a conventional alphanumeric input devicefor communicating information between computer system 900 and computeroperator(s). Another type of user input device is cursor control device923, such as a mouse, touch mouse, trackball, or other type of cursorfor communicating information between system 900 and user(s).

Communication device 925 is coupled to bus 912 for accessing informationfrom remote computers or servers through wide-area network.Communication device 925 may include a modem, a router, or a networkinterface device, or other similar devices that facilitate communicationbetween computer 900 and the network. In one aspect, communicationdevice 925 is configured to perform wireless functions.

In one embodiment, WCC 930 is coupled to bus 910 and is configured toprovide sub-block Cholesky decomposition to generate whiteningcoefficients with which to filter received data to aid in detection. Invarious exemplary embodiments, the WCC 930 comprises hardware, firmware,or a combination of hardware, and firmware. For example, in oneembodiment, the WCC 930 comprises the whitening coefficient calculator700 shown in FIG. 7. In an exemplary embodiment, the WCC 930 andcommunication device 925 perform data reception and whitening inaccordance with one embodiment of the present invention.

While particular embodiments of the present invention have been shownand described, it will be obvious to those skilled in the art that,based upon the teachings herein, changes and modifications may be madewithout departing from this exemplary embodiment(s) of the presentinvention and its broader aspects. Therefore, the appended claims areintended to encompass within their scope all such changes andmodifications as are within the true spirit and scope of this exemplaryembodiment(s) of the present invention.

What is claimed is:
 1. A method for parallel processing sub-blockmatrices (R₀₀, R₁₀, and R₁₁) of a covariance matrix (R) to determine awhitening coefficient matrix (W), the method comprising: calculating afirst whitening matrix W₀₀, lower triangle matrix L₀₀, and diagonalmatrix D₀₀ from the sub-block matrix R₀₀; calculating a lower trianglematrix L₁₀ from the sub-block matrix R₁₀ and the matrices L₀₀ and D₀₀;calculating a matrix X from the matrices D₀₀ and L₁₀; calculating amatrix Z from the matrix X and the sub-block matrix R₁₁; calculating athird whitening matrix W₁₁, lower triangle matrix L₁₁, and a diagonalmatrix D₁₁ from the matrix Z; and calculating a second whitening matrixW₁₀ from the matrices L₀₀, L₁₀, L₁₁, and D₁₁.
 2. The method of claim 1,wherein each of the operations of calculating the first whitening matrixand calculating the second whitening matrix comprise receiving aHermitian input matrix and calculating elements associated with selecteddiagonal, lower triangle, and whitening matrices.
 3. The method of claim1, wherein each of the operations of calculating the first whiteningmatrix and calculating the second whitening matrix comprises calculating(d₀₀, d₁₁) of the selected diagonal matrix and (l₁₀) of the select lowertriangle matrix according to: d₀₀ = r₀₀$d_{11} = {{r_{11} - \frac{{r_{10}}^{2}}{r_{00}}} = {r_{11} - \left( \frac{{r_{10\mspace{11mu} {Re}}*r_{10\mspace{11mu} {Re}}} + {r_{10\; {Im}}*{r_{10\mspace{11mu} {Im}}}_{\;}}}{r_{00}} \right)}}$$l_{10} = {\frac{r_{10}}{r_{00}} = \frac{r_{10\mspace{11mu} {Re}} + {jr}_{10\mspace{11mu} {Im}}}{r_{00}}}$4. The method of claim 1, wherein each of the operations of calculatingthe first whitening matrix and calculating the second whitening matrixcomprises calculating (d₀₀, d₁₁) of the selected diagonal matrix and(l₁₀) of the select lower triangle matrix according to: d₀₀ = r₀₀$l_{10} = {\frac{r_{10 \cdot {Re}}}{r_{00}} + {j\frac{r_{10 \cdot {Im}}}{r_{00}}}}$$d_{11} = {r_{11} - \left( {{\frac{r_{10 \cdot {Re}}}{r_{00}}*r_{10 \cdot {Re}}} + {\frac{r_{10 \cdot {Im}}}{r_{00}}*r_{10 \cdot {Im}}}} \right)}$5. The method of claim 1, wherein each of the means for calculating thefirst whitening matrix and the means for calculating the secondwhitening matrix comprises calculating (W₀₀, W₁₀, W₁₁) of the selectedwhitening matrix according to:W ₀₀=1/sqrt(d ₀₀)W ₀₁=0W ₁₀ =−−l ₁₀/sqrt(d ₁₁)W ₁₁=1/sqrt(d ₁₁)
 6. An apparatus that parallel processes sub-blockmatrices (R₀₀, R₁₀, and R₁₁) of a covariance matrix (R) to determine awhitening coefficient matrix (W), the apparatus comprising: a first LDLcoefficient calculator that calculates a first whitening matrix W₀₀,lower triangle matrix L₀₀, and diagonal matrix D₀₀ from the sub-blockmatrix R₀₀; a first matrix calculator that calculates a lower trianglematrix L₁₀ from the sub-block matrix R₁₀ and the matrices L₀₀ and D₀₀; asecond matrix calculator that calculates a matrix X from the matricesD₀₀ and L₁₀; a matrix subtractor that calculates a matrix Z from thematrix X and the sub-block matrix R₁₁; a second LDL coefficientcalculator that calculates a third whitening matrix W₁₁, lower trianglematrix L₁₁, and a diagonal matrix D₁₁ from the matrix Z; and a thirdmatrix calculator that calculates a second whitening matrix W₁₀ from thematrices L₀₀, L₁₀, L₁₁, and D₁₁.
 7. The apparatus of claim 6, furthercomprising an output combiner that combines the first, second, and thirdwhitening matrices to form a whitening coefficient matrix that isoutput.
 8. The apparatus of claim 6, wherein the first matrix calculatorcomprises an inverter and a multiplier, and wherein the inverterreceived the matrices L00 and D00 and determines an output matrix from(D₀₀L₀₀ ^(H))⁻¹ and the multiplier multiples the output matrix with thesub-block matrix R₁₀ to calculate L₁₀.
 9. The apparatus of claim 6,wherein the second matrix calculator calculates the matrix X from(L₀₀D₀₀L₁₀ ^(H)).
 10. The apparatus of claim 6, wherein the matrixsubtractor subtracts the matrix X from the sub-block matrix R₁₁ todetermine the matrix Z.
 11. The apparatus of claim 6, wherein the thirdmatrix calculator calculates W₁₀ from −D₁₁ ^(−1/2)(L₁₁ ⁻¹L₁₀L₀₀ ⁻¹). 12.The apparatus of claim 6, wherein each of the first and second LDLcoefficient calculators is configured to receive a Hermitian inputmatrix and calculate elements associated with selected diagonal, lowertriangle, and whitening matrices.
 13. The apparatus of claim 12, whereineach of the LDL coefficient calculators comprises a D/L calculator thatincludes five multipliers, two adders and one (1/x) function thatcalculate (d₀₀, d₁₁) of the selected diagonal matrix and (i_(10.RE),i_(10.IM)) of the select lower triangle matrix according to: d₀₀ = r₀₀$d_{11} = {{r_{11} - \frac{{r_{10}}^{2}}{r_{00}}} = {r_{11} - \left( \frac{{r_{10{Re}}*r_{10{Re}}} + {r_{10{Im}}*r_{10{Im}}}}{r_{00}} \right)}}$$I_{10} = {\frac{r_{10}}{r_{00}} = \frac{r_{10{Re}} + {fr}_{10{Im}}}{r_{00}}}$14. The apparatus of claim 12, wherein each of the LDL coefficientcalculators comprises a D/L calculator that includes four multipliers,two adders and one (1/x) function that calculate (d₀₀, d₁₁) of theselected diagonal matrix and (i_(10.RE), i_(10.IM)) of the select lowertriangle matrix according to: d₀₀ = r₀₀$l_{10} = {\frac{r_{10 \cdot {Re}}}{r_{00}} + {j\frac{r_{10 \cdot {Im}}}{r_{00}}}}$$d_{11} = {r_{11} - \left( {{\frac{r_{10 \cdot {Re}}}{r_{00}}*r_{10 \cdot {Re}}} + {\frac{r_{10 \cdot {Im}}}{r_{00}}*r_{10 \cdot {Im}}}} \right)}$15. The apparatus of claim 12, wherein each of the LDL coefficientcalculators comprises a coefficient calculator that includes twomultipliers and two (1/x) functions that calculate (W_(00.Re),W_(10.Re), W_(10.im), W_(11.Re)) of the selected whitening matrixaccording to:W ₀₀=1/sqrt(d ₀₀)W ₀₁=0W ₁₀ =−l ₁₀/sqrt(d ₁₁)W ₁₁=1/sqrt(d ₁₁)
 16. An apparatus that parallel processes sub-blockmatrices (R₀₀, R₁₀, and R₁₁) of a covariance matrix (R) to determine awhitening coefficient matrix (W), the apparatus comprising: means forcalculating a first whitening matrix W₀₀, lower triangle matrix L₀₀, anddiagonal matrix D₀₀ from the sub-block matrix R₀₀; means for calculatinga lower triangle matrix L₁₀ from the sub-block matrix R₁₀ and thematrices L₀₀ and D₀₀; means for calculating a matrix X from the matricesD₀₀ and L₁₀; means for calculating a matrix Z from the matrix X and thesub-block matrix R₁₁; means for calculating a third whitening matrixW₁₁, lower triangle matrix L₁₁, and a diagonal matrix D₁₁ from thematrix Z; and means for calculating a second whitening matrix W₁₀ fromthe matrices L₀₀, L₁₀, L₁₁, and D₁₁.
 17. The apparatus of claim 16,wherein each of the means for calculating the first whitening matrix andthe means for calculating the second whitening matrix is configured toreceive a Hermitian input matrix and calculate elements associated withselected diagonal, lower triangle, and whitening matrices.
 18. Theapparatus of claim 16, wherein each of the means for calculating thefirst whitening matrix and the means for calculating the secondwhitening matrix comprises a D/L calculator that includes fivemultipliers, two adders and one (1/x) function that calculate (d₀₀, d₁₁)of the selected diagonal matrix and (i_(10.RE), i_(10.IM)) of the selectlower triangle matrix according to: d₀₀ = r₀₀$d_{11} = {{r_{11} - \frac{{r_{10}}^{2}}{r_{00}}} = {r_{11} - \left( \frac{{r_{10 \cdot {Re}}*r_{10 \cdot {Re}}} + {r_{10 \cdot {Im}}*r_{10 \cdot {Im}}}}{r_{00}} \right)}}$$l_{10} = {\frac{r_{10}}{r_{00}} = \frac{r_{10 \cdot {Re}} + {jr}_{10 \cdot {Im}}}{r_{00}}}$19. The apparatus of claim 16, wherein each of the means for calculatingthe first whitening matrix and the means for calculating the secondwhitening matrix comprises a D/L calculator that includes fourmultipliers, two adders and one (1/x) function that calculate (d₀₀, d₁₁)of the selected diagonal matrix and (i_(10.RE), i_(10.IM)) of the selectlower triangle matrix according to: d₀₀ = r₀₀$l_{10} = {\frac{r_{10 \cdot {Re}}}{r_{00}} + {j\frac{r_{10 \cdot {Im}}}{r_{00}}}}$$d_{11} = {r_{11} - \left( {{\frac{r_{10 \cdot {Re}}}{r_{00}}*r_{10 \cdot {Re}}} + {\frac{r_{10 \cdot {Im}}}{r_{00}}*r_{10 \cdot {Im}}}} \right)}$20. The apparatus of claim 16, wherein each of the means for calculatingthe first whitening matrix and the means for calculating the secondwhitening matrix comprises a coefficient calculator that includes twomultipliers and two (1/x) functions that calculate (W₀₀, W₁₀, W₁₁) ofthe selected whitening matrix according to:W ₀₀=1/sqrt(d ₀₀)W ₀₁=0W ₁₀ =−l ₁₀/sqrt(d ₁₁)W ₁₁=1/sqrt(d ₁₁)